New wave behaviors of the Fokas-Lenells model using three integration techniques

In this investigation, we apply the improved Kudryashov, the novel Kudryashov, and the unified methods to demonstrate new wave behaviors of the Fokas-Lenells nonlinear waveform arising in birefringent fibers. Through the application of these techniques, we obtain numerous previously unreported novel dynamic optical soliton solutions in mixed hyperbolic, trigonometric, and rational forms of the governing model. These solutions encompass periodic waves with W-shaped profiles, gradually increasing amplitudes, rapidly increasing amplitudes, double-periodic waves, and breather waves with symmetrical or asymmetrical amplitudes. Singular solitons with single and multiple breather waves are also derived. Based on these findings, we can say that our implemented methods are more reliable and useful when retrieving optical soliton results for complicated nonlinear systems. Various potential features of the derived solutions are presented graphically.

A renowned model, known as the Fokas-Lenells model, was first presented in 2009 [28] and has since achieved various honors.There have been numerous applications for this model, including fiber optics.The mentioned system can be used to describe the dynamic features of optical and photonic crystal fibers [29].To express soliton solutions, the Fokas-Lenells PDE has been studied by applying numerous reliable and effective elegant algorithms, such as the Ricati equation scheme [30], extended trial equation scheme [31], complex envelope function ansatz [29], mapping scheme [32], unified solver method [33], φ 6 -model expansion approach [34], etc.
The prime focus of this research is to present the novel wave behaviors of the suggested model using three integration schemes, the unified method [35,36], the improved Kudryashov method [37], and the novel Kudryashov technique [38,39] which can describe the dynamic feature of the Fokas-Lenells model.
This research literature is formulated as underneath: Section 2 contains the operating model.An ordinary differential form of the operating model can be found in Section 3. Sections 4, 5, and 6 summarize and implement the unified method, the improved Kudryashov method, and the novel Kudryashov technique, respectively.Section 7 describes the graphic analysis with a discussion of the operating model's solutions.Finally, a summary of the article and plans for future research are given in Section 8.
We confirm that the optical soliton results of the Fokas-Lenells dynamical waveform obtained through our employed methods are the first reported and have not been studied until now.

Operating model
The dimensionless Fokas-Lenells PDE has the following form [28][29][30][31][32][33][34]: In the aforementioned model, Q(x, t) denotes the wave's magnitude with distance coordinate x and time coordinate t, where i ¼ ffi ffi ffi ffi ffi ffiffi À 1 p .The potentials a, a 1 , a 2 , and s signify the coefficient of inter-model dispersion, GVD, STD, and nonlinear dispersion, sequentially.The potentials b, n, and l signify self-phase modulation, the effect of full non-linearity, and the selfsteepening effect, sequentially.Finally, m is another type of nonlinear dispersion.It's important to note that the parameter m holds a real numerical value.Nonetheless, if m is entirely imaginary, it would depict Raman scattering.This phenomenon contributes to the frequency shift of solitons and is characterized by the dissipative Raman effect.We can say that, the Fokas-Lenells model is a comprehensive equation that combines dispersion, nonlinearity, and various effects to describe the evolution of a wave's magnitude Q(x, t).The coefficients a, a 1 , a 2 , s, m, b, n, and l play distinct roles in capturing the influence of different physical phenomena on the wave's behavior.

ODE formulation of the model
To resolve equation Eq (2.1), we will take into account the following solution structure: ), the imaginary and real portions will be separated.Then the real part is and, the imaginary portion implies From Eq (3. 3), we have In Eq (3.4) by plugging [2mn + l(2n + 1)]U 2n = 0, s = 0, one reaches Accordingly, from Eq (3.2) we have
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where δ = −kx + wt+ p. Applying Eq (3.1) and Eqs (4.3)-(4.5),by the aid of solution Eq (4.8) gives the next 8 exact solutions of Eq (2.1).
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where δ = −kx + wt + p. Applying Eq (3.1) and Eqs (4.3)-(4.5),by the aid of solution Eq (4.9) gives the next 8 exact solutions of Eq (2.1).
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where δ = −kx + wt + p.

Description of the improved Kudryashov method with application
Let the auxiliary solution of the suggested nonlinear structure as follows [37] UðBÞ For finding N in Eq (5.1), by balancing U 3 and U 00 yields N = K + 1.If K = 1, then N = 2 and Eq (5.1) can be given in the formation Now using Eqs (5.6), (5.2), and (3.6) and some simple calculation gives ; l 0 ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2 p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ba ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ð5:9Þ Applying Eqs (5.3)-(5.5)and Eq (3.1), by the aid of the solution Eq (5.7) gives the next 5 exact solutions Eq (2.1).

Description of the novel Kudryashov method with application
The auxiliary solution to the suggested nonlinear structure is assumed in the underneath symbolic form [38,39] UðBÞ ¼ C 0 ðBÞ ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi CðBÞ 2 ð1 À 4LMCðBÞ 2 Þ q ; ð6:2Þ From equation Eq (6. 2), we can obtain the remarkable relationship as follows: with real potentials L, and M. For finding K in Eq (6.1), balance between U 3 and U 00 yields K = 1.As a result Eq (6.1) will be converted as Now, making use of Eqs (6.3), (6.4), and (3.6) and some simple calculation reads According to Eqs (6.4) and (6.5), the upcoming result of the mentioned model is obtained: where δ = −kx + wt + p with w ¼

Figure analysis with discussion
Solutions Q 26 and Q 27 produce W-shaped periodic waves.We depict only Q 26 (see Figs 1 and  2) for and Q 25 display a single periodic wave with properties that can either amplify or reduce the wave's amplitude.We depict only Q 20 (see  for We can see that (see Figs 3 and 4) if b = 1, the amplitudes of waves rise over time, whereas if b = −1, the opposite phenomenon occurs (see Figs 5 and 6).Solutions Q 10 , Q 11 , and Q 18 display periodic waves with rapidly increasing or decreasing amplitudes.We depict only Q 10 (see Figs 7-9) for a 1 = a 2 = 4, k = 5, ϑ = −1, b = m 1 = G = H = ρ = p = 1.We can see that (see Fig 7) if x = −5, the amplitudes of waves quickly increase over time, whereas if x = 3, the opposite phenomenon occurs (see Fig 9).Furthermore, for x = −1, the wave's amplitudes rapidly increase and decrease over time (see Fig 8).Solutions Q 4 and Q 36 correspond to double periodic waves, which are depicted by Q 4 (see  for By employing the unified technique, the improved Kudryashov scheme, and the novel Kudryashov approach, the present analysis of the Fokas-Lenells model recovers various novel waveforms and obtains some earlier results when we compare our results to references therein [40,41].Our suggested methods are versatile enough to yield solutions to various nonlinear wave equations in hyperbolic, trigonometric, periodic, and exponential forms.Importantly, these solutions can be ready for use without being reduced to any other form.Consequently, there is no necessity to further transform the solutions into other representations.Various dynamic characteristics of outcomes are displayed in 3D, 2D, and density diagrams by setting the parameters involved.The correctness of the calculations was confirmed by reintegrating them into the governing model after wave profiles were created using Maple 18.

Conclusion
The unified, the improved Kudryashov, and the novel Kudryashov schemes are successfully used in this manuscript to find new waveforms for the Fokas-Lenells dynamical form.The resulting waveforms include periodic W-shaped waves (refer to Figs 1 and 2), periodic waves with gradually increasing amplitudes (observe Figs 3-6), rapidly increasing amplitudes (view Figs 7-9), and double-periodic waves (see .Single-breather waves (see Figs 14-17) and multi-breather waves with symmetric (see  and asymmetric (see  amplitudes are also obtained.Additionally, we obtain singular solutions with single (see Figs [26][27][28][29] and multibreather waves (see .These findings demonstrate that our employed methods are more useful and reliable tools to retrieve optical soliton outcomes for complicated nonlinear models.
Qðx; tÞ ¼ UðBÞ exp ðidÞ; ð3:1Þ whilst B = x − gt with velocity component g in which the phase component δ = −kx + wt + p and the amplitude component U whereas wave number w, frequency k, and phase value p.Now by utilizing Eqs (2.1) and (3.1